prodibunt aequationes,

=

(ax+ By) - Dy2= aq, (a'x' + By') - Dy" a'q' ((ax+ By) (a'x' + By')+ Dyy')2-D((ax+By)y'+(a'x'+By')y)2 = aa'44'. Si iam observamus, propter D= B2 — aa'C, haberi

(4) (ax+ By) (a'x' + By') + Dyy'

=

aa'X+BY

ubi positum est

απ' φφ'.

(5) X=xx'—Cyy', Y=(ax+By)y'+(a'x'+By')y=axy'+a'x'y+2Byy', aequatio ultima, per aa' divisa, induet formam

eiusdem de

x', y valores

m, y' = m'

Postquam productum formarum et ', per formam terminantis D indefinite exhibuimus, tum ipsis x, y tum ipsis determinatos inter se primos tribui supponamus, pro quibus evadat, et qui conditionibus supra indicatis satisfaciant. Quo facto, demonstrandum erit 1°. repraesentationem ipsius mm' per formam 4,. quam aequationibus (5) et (6) praeberi videmus, fore propriam, sive X et Y a divisore communi fore liberos et 2°. radicem ad quam haec repraesentatio pertinet et quae sit (mm', Z), revera ex radicibus (m, 5), (m', ') ad quas repraesentationes ipsorum m et m' pertinere supponemus, fore compositam.

1o. Ut evincatur, ipsos X et Y divisorem communem non habere, proficiscamur a suppositione, numerum primum p ambos metiri et quae hinc sequantur videamus. Quum p alterutrum ipsorum m, m' metiri debeat, ponamus id quod licet, m ipsius p esse multiplum. Iam dico, etiam ipsum m' per p divisibilem esse supponendum. Quum enim X et Y per p sint divisibiles, patet, multiplicando aequat. (5) per —ay' et x', et addendo, proditurum esse integrum per p divisibilem (a'x"2 + 2 Bx'y' +aCy") y = m'y. Si iam ipsum n' per p divisibilem non esse supponere velis, ipse y et proin etiam ipsius p erunt multipla. Tum autem propter xx' - Cyy' = X=0 (mod p), et ex eo quod ad ipsum y est primus, colligitur x'=0, unde tandem m'u'x"2 + 2Bx'y' +aCy" ipsius p esse multiplum perspicitur. Quum p ambos m et m' metiatur, erit per aequat. (1), ax + By = −y, a'x' + By' =y's' (mod p), quarum formularum substitutione, secunda aequat. (5) transit in congr. (+)yy' = 0 (mod p). Si +0 esset, sequeretur S'=-S contra hyp., radices et inter se concordare sive' esse. Restat ut videamus quid ex alterutra suppositionum y = 0, y' 0, quae plane inter se

=

12

sunt similes, sequatur. Si y ideoque propter xx' - Cyy' = 0, etiam ' per p esset divisibilis, haberemus per aequat. (2) et (3) resp., ?=B, ['=—B, et perinde ut supra, += 0 (modp). Evictum est igitur, ipsos X et Y

inter se esse primos.

(-Sy'a'x'+By')y=Y,

2o. Ut iam probemus, radicem ad quam repraesentatio ipsius mm' pertinet et quam per (mm', Z) designavimus revera ex ipsis (m, 5), (m', [') esse compositam, propter symmetriam ostendere sufficiet, esse Z= respectu cuiuslibet divisoris primi p ipsius m. Quum per aeq. (1) habeatur ax + By =-Sy (modp), ex aeq. (4) et secunda aeq. (5) facile deducuntur congruentiae (-(ax+By') + Dy')yaa'X+BY, unde, posteriore per multiplicata ad priorem addita, et ratione habita formulae D, sequitur aa'X÷BY: Y (mod p) qua congr. cum hac aa' X + BY = — YZ (mod p) comparata, vides esse Z=5 (mod p), si ipsum Y non metiatur. Restat ut consideremus casum ubi Y per p est divisibilis, quo casu erit per aeq. (2), Z= B. Si iam etiam y ipsius p est multiplum, erit simili modo, B et proinde Z. Si vero y per p non est divisibilis, e congr. quam supra nacti eramus, concluditur esse a'x' + By' —¿y' = 0, unde facile deducitur, ipsum p producti a'm' esse factorem. Est enim a'm' = (a'x' + By')' — Dy"2 = (a'x' +- By')' — 5y0. Iam duo casus sunt distinguendi. Supponamus primo, ipsum a' per p non esse divisibilem. Quo casu quum m' per p sit divisibilis, habetur a'x' +- By' + S'y' =0, qua formula cum superiore collata sequitur (+)'=0, quae congr. ad conclusionem absurdam deducit. Nam quum + multiplum ipsius p esse non posS sit, sequeretur y'=0, ideoque propter m'a'x'2 + 2Bx'y' + aCy”, a'=0 contra hyp. Quare hic casus locum habere non potest. In casu posteriori, ubi a' per p est divisibilis, e congr. a'x' + By'—¿y'=0, deducitur (B—S)y'=0. Si iam p ipsum y' non metitur, habetur (=B, id quod cum congr. Z=B (mod p) convenit. Si vero p ipsius y' ideoque etiam ipsius m'est divisor, habetur per aeq. (2), S'=B, unde ut supra esse B colliges, si radices Set inter se concordantes esse attenderis.

-

10.

An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism.

(By the late George Green, fellow of Gonville and Cains - Colleges at Cambridge.) *)

Application of the preceding results to the theory of electricity.

8.

The first application we shall make of the foregoing principles, will

be to the theory of the Leyden phial. For this, we will call the inner surface of the phial A, and suppose it to be of any form whatever, plane or curved, then, B being its outer surface, and the thickness of the glass measured along a normal to A; will be a very small quantity, which, for greater generality, we will suppose to vary in any way, in passing from one point of the surface A to another. If now the inner coating of the phial be put in communication with a conductor C, charged with any quantity of electricity, and the outer one be also made to communicate with another conducting body C', containing any other quantity of electricity, it is evident, in consequence of the communications here established, that the total potential function, arising from the whole system, will be constant throughout the interior of the inner metallic coating, and of the body C. We shall here represent this constant quantity by

B.

Moreover, the same potential function within the substance of the outer coating, and in the interior of the conductor C', will be equal to another constant quantity B'.

Then designating by V, the value of this function, for the whole of the space exterior to the conducting bodies of the system, and consequently for that within the substance of the glass itself; we shall have (art. 4)

One horizontal line over any quantity, indicating that it belongs to the inner surface A; and two showing that it belongs to the outer one B.

*) Vide tome 39 p. 13 and tome 44 p. 356 of this Journal.

At any point of the surface A, suppose a normal to it to be drawn, and let this be the axes of : then ', ", being two other rectangular axes, which are necessarily in the plane tangent to A at this point; V may be considered as a function of w, w' and w", and we shall have by Taylor's theorem, since 0 and w w = w" =0 at the axis of w along which is measured,

where, on account of the smallness of 0, the series converges very rapidly.

By writing in the above, for V and V their values just given, we obtain

In the same way, if w be a normal to B, directed towards A, and 0, be the thickness of the glass measured along this normal, we shall have

But, if we neglect quantities of the order 0, compared with those retained, the following equation will evidently hold good,

n being any whole positive number, the factor (-1)" being introduced because u and w are measured in opposite directions. Now by article 4

and being the densities of the electric fluid at the surfaces A and B respectively. Permitting ourselves, in what follows, to neglect quantities of the order compared with those retained, it is clear that we may write 0 for 0, and hence by substitution

where and are quantities of the order; ' and ß being the ordre Ø

or unity. The only thing which now remains to be determined, is the value d2V

of for any point on the surface A.

dw2

0=

Throughout the substance of the glass, the potential function V will satisfy the equation 0 V, and therefore at a point on the surface of A, where of necessity, w, w', and w", are each equal to zero, we have

the horizontal mark over w, w' and w" being, for simplicity, ommitted. Then since w' = 0,

and as is constant and equal to at the surface A, there hence arises

R being the radius of curvature of the surface 4, in the plane (w, w'). Substituting these values in the expression immediately preceding, we get

In precisely the same way we obtain, by writing R' for the radius of curvature in the plane (w, w"),

both rays being accounted positive on the side where w, i. e. w is negative. These values substituted in 0d, there results